Recent scientific research suggests there may be surprises in store in the distribution tails of catastrophes. Dickie Whitaker and Gordon Woo explain.
Catastrophe reinsurers deal with extreme risks which are defined statistically by the tails of distributions. Standard actuarial methods have evolved over the years to quantify these tails, but are there surprises in store? Recent scientific research suggests that there may be.
Having a sense of what is normal, and what is abnormal, is essential for the efficient operation of any commercial business. A manager of a shoe shop, for example, has to decide on the distribution of shoe sizes to stock and make immediately available to meet customer demand. To ask for a pair of stylish shoes, only to find one's size is not stocked, is a common frustration for unusually short or tall people. To minimise customer dissatisfaction, a stock manager needs an idea of the distribution of shoe sizes within the adult population served by the shop. Without undertaking an actual survey to establish this statistical distribution, it may be reasonable for him to assume that this is normal, i.e. follows a bell-shaped curve.
Normal distribution, so-called because of its widespread applicability, is also referred to as Gaussian, after the German genius who first discovered it as a law of measurement errors. For the normal distribution, the chance of a value exceeding six standard deviations from the mean is minuscule: one-in-a-billion. A manager of a large shop could not be criticised for failing to outfit a customer who figured in the Guinness Book of Records.
However, he might hope to cater for the exceptional person in 5000, perhaps a professional athlete like the New Zealand All-Black rugby star, JonahLomu, with size 15 feet.
The significance of the comparison between odds of one-in-a-billion and one-in-5000, goes well beyond footwear: it concerns all reinsurers, (irrespective of their personal physical attributes). An article in the Financial Times, drew attention to recent scientific research showing observational evidence for unexpected departures from the normal distribution. Instead of the familiar bell-shaped curve, another type of curve achieves prominence: one looking more like a coolie's straw hat than a church bell. The key feature of the new curve is that the probability of a rare random fluctuation is much greater than for the bell-shaped curve.At six standard deviations from the mean, this probability may be as high as one-in-5000, as opposed to one-in-a-billion for the bell-shaped curve.
Shopkeepers need not be alarmed by this research: there will not be a sudden rush for size 20 shoes, and, in any case, lack of preparedness would not force them to shut down. But reinsurers of risks from natural perils should take note. Associated with each hazard event is a geographical map of hazard severity - known metaphorically as a footprint. For a hurricane, this is the area of damaging wind speeds; for an earthquake, this is the area of damaging ground shaking and for a flood, the area of inundation. Any reinsurer who assumed a normal distribution for these footprints would be courting insolvency. Large footprints occur far more often than would be surmised from a normal distribution.
The scientific reasons for the more frequent occurrence of large events are becoming clear. At the core of the scientific explanation is the concept of self-similarity: hazard phenomena have the same general appearance across a broad range of spatial scales. Thus, a close-up photograph of a rock fracture usually affords little clue as to its absolute dimension, which might be measured in millimetres or metres. Crucially, this feature of self-similarity governs the scale range of natural hazards, which extends over orders of magnitude in variation of spatial extent and energy release. Those who find it hard to appreciate this scale range, may implicitly be using the normal distribution as a default, perhaps subconscious, yardstick for natural variation. Past loss experience will always serve as a practical guide to underwriters, but no amount of individual experience is sufficient to resolve adequately the tail of a catastrophe loss distribution.Mathematical formulae based on the fundamental concept of self-similarity do form an integral part of the risk engines which drive catastrophe models. Portfolio loss estimates at long return periods are much higher than would be the case if the normal distribution were used in hazard models. But catastrophe modellers can never afford to be complacent over their assumptions, especially where they concern possible correlations between hazard phenomena. Mindful of the spate of damaging earthquakes in the latter half of 1999, reinsurers will be wary of variability in event frequency as well as severity. Is the tail of the global distribution of earthquakes heavier than anticipated by assuming the statistical independence of random fluctuations in seismicity?
This is a basic seismological question, the answer to which may best be sought from within the twin scientific fields of mathematics and physics, which underlie the new research by Professor Turcotte of Cornell University and co-workers. These fields have contributed much, and have yet more to offer in the improvement of catastrophe models. However, with most mathematical physicists drawn into back-room derivatives analysis, these fields are sparsely represented within the community of catastrophe modellers, which is far stronger in the more practical engineering andeconomic disciplines.
Dickie Whitaker is director of Intermediary Systems Ltd. (ISL), subsidiary of Guy Carpenter. Tel +44 (0) 207 357 2251; e-mail firstname.lastname@example.org and Dr Gordon Woo is a consultant with the firm. Dr Woo is the author of The Mathematics of Natural Catastrophes, published by Imperial College Press, which gives insight into the quantitative basis for catastrophe modelling.